 Ie, I guess we should... We should get going. So we have on... we have today to discuss some... ...unfortunatly, rather formal stuff... ...and tomorrow we will do something that's physically more interesting... ...the Einstein-Pedalski-Rosn experiment... ...but which will draw heavily on what we're going to do today. Mae'n gwneud yn benedig allu'n mwyaf o'r suturdau a allu'r hydrodinatumau yn bryd, fel ychydig, mae hydrodinatum yn ymweld eich hydrodinatun, a'r proton. Mae'n gilydd o'r hydrodinatum, mae'n gilydd o'r hydrodinatum, a'n gilydd o'r hydrodinatun o'r proton. Mae'n deimond yn yma, gan y gydig cadw, mae'n fwy o dmitraen ddeimARS. Which a diamond would contain 10 to the 23 or whatever carbon atoms and to know the state of the diamond. Officially, you would need to know the state of the 10 to the 23 carbon atoms. So it's going to be important to move forward towards applying quantum mechanics to any nontrivial and really interesting system, we're going to have to learn how to describe systems that come in parts. ac mae'r gael Dodhaidau yn gweinio gwasanaeth... ..werth o wahanol, ac mae'n cymhwyl iawn... ..y holl Downeydd. Dyma'r unig i gyda hybridd. Ac mae'n cymhwyl iawn, yw'r cyllid ganfyrdd. I'r cyllid gyda hybriddynd gan'r cymhwyl iawn. Mae'r cyllid gyda hynny... ..fennyddio pand yna o'r gyllid gyda hybridd... ..y maen nhw ychydigwyd gan gyllid gyda hynny. Mae'r cyllid gyda hynny i gychydigwyd. or whatever. So there are springs, as it were. There are things connecting the different parts, but it turns out that the quantum mechanics of a system made up of two objects is non-trivally... is non-trivally... stri Îvally different from the quantum mechanics of the two isolated things, even if you just logically consider them to be the same. So when we do angular momentum, in the coming weeks we will find very strange and interesting results ddwy gennych y tu gyfnod o'r cwrs sydd wedi'i gweithio'r gwrs. Gweithio'r chweithio y cwrs yn y bach. Dwi'n gwneud o'r cwrs yma o'r cwrs yma o'r cyfnod o'r gwych ar gyfer y gwrs a'i gymuned yna'r ddweud o'r cyfnod o'r cyfnod. Felly, roedd y cyfnod y Gweithgafn Gweithgafn Gweithgafn yn y bach yn dengau'r cyfnod o'r cyfnod o'r cyfnod o'r gyfnod. So we are talking about today, is putting things logically together to make compound systems and there may or may not be springs connecting, physically connecting these things. The central problem of the central thing we have to address is if we have system A and system B so we have two distinct systems A yna, nes i ddweud ystod, mae'n iawn. Rwy'n gwneud y systyn yn ystod, ac mae'n mynd i'r sylwf yn ystod, ac mae'r ddiweddau sy'n gweithio'n gwneud ystod o'r sylwf yn sylw ffrindiau. Rwy'n gwneud ystod. A rwy'n eu ddweud yn ddweud ystod. A dyna ystyried ystod i'r sylw, ystod ystod o'r sylw ystod i'r sylw ystod, The system that you get by considering A and B together. So this might be the electron, this might be the proton, what's the state of the compound system which we call a hydrogen atom, for example. Cos if we know how to add, if we know how to compound A, system A with system B, to make a combined system, we can compound another one, we can use the same rule to add another element of a bigger system, A, B, C, and so on and so forth, and eventually we can build up a diamond dan 10-23 carbon atoms. The central, once you know how to add two systems by repeating this process adding more and more systems you can put any number of systems together. So this is the central problem that we have to address. So the states... First of all, we just write some formal stuff. The states of the compound system AB, this is when you logically think of the system at A with B That's one system are of well some of the states of the system a b are written like this a b semicolon i j. And we write it symbolically as a semicolon, ups i b semicolon j, sorry, the semiconductor on the j look too similar. Yn dweud, yna'r argymall ac mae'r wyngarol yn ddweud ac mae'r ddweud yn ei bodan bod yn linio'r ddweud ac mae'n ddweud, os ystod y celfm ar broses. Mae'r ddweud yn ddweud, hwnnw, yn ei ddweud, ac mae'r ddweud, hwnnw, yn ei ddweud. sefyd give meaning to that. On the right hand side, we have a symbolic multiplication and we don't need to worry too much. You'll see as we go on that we don't need to worry too much about what exactly we mean by this multiplication. But this is just a symbolic product of... Efficially, it's a tencer product, but we don't want to frighten everybody. This is a symbolic multiplication of a ket on a ket. We'll find out how to interpret that as we go along. Then obviously we can have the bras. There must be associated bras. Since this is a state of a system, it has a bra, which will be i, j, oops, is equal to, of course, the logical product of the bras. We give meaning to this thing by explaining what happens when this goes on to this. When this goes on to this, we should get a complex number. To give meaning to all this, I need to explain what this is. i', j' on a, b, a, b, semicolon i, j is equal to. This should be a complex number. To give meaning to all this hocus pocus, I need to explain which complex number. The complex number it is, is this complex number. a i' a i times bj' bj. What I've written on the right makes perfectly, is completely well defined because this is a complex number and this is a complex number and we can multiply complex numbers. We get a complex number which gives meaning to this on the left. That's really, really, really the essence of giving meaning to this thing here because remember we only want these kets or we want with kets in order to calculate amplitudes, whose mod squares are going to be the probabilities for, give us our predictions. As long as you know how to get an amplitude out of a ket you know enough about the ket to get on with it. We've given meaning to the process by which we extract amplitudes out of kets which is this brine through business. That leads to the experimental predictions which are the whole point of the theory. Why does this make sense? Why is this a sensible definition? This is the definition of what we mean by these animals. Why is it a sensible definition? Well, it says that the probability of getting, of measuring the results i primed and j primed given that we're in this state is equal to obviously the mod square of this horrible thing a, b, i primed, j primed, a, b, i, j mod square. That's how we would interpret this complex number and according to this formula this is equal to the product of the probability associated this is the probability for the system a, b the probability system a of getting the result i primed times the probability of getting the result j primed. Because if I take the mod square of both sides the mod square of this product is the product of the mod squares the mod squares on this side by definition are these probabilities. So the probability, this says the probability that if I take measurements of my combined system I find that a is in the i primed state and j is in the b is in the j primed state is simply the product of the probabilities that the a system is in the i primed state and the b system is in the j primed state if you make individual measurements. So that makes perfect sense. And it's motivating. This is why we write the catch of the compound system like this this multiplication rule, this symbolic multiplication is inherited from this law for multiplying probabilities in probability theory. Now that having said that and everything's nice and simple we have to make the point that I now want to show that not all states this is the thing that's surprising of a, b are of the form a, b. That's what I want to now establish that this is that it's not true that all states of the system are of this form. Okay, so let's so for example consider two state systems. So we're going to do a concrete example to illustrate this general and very fundamental principle. We're going to have two two state systems we're going to have a is going to have states plus and minus. These are complete set. So we're considering the simplest non-trivial example and b is going to have the states up and down. This is just notation that enables us to by using a plus sign and a minus sign for a up arrow and a down arrow for b I avoid the necessity of writing down these pesky a, b labels. Let's now consider let the state of a b a plus plus a minus minus. So this is a general state. By taking a linear combination of the two basis vectors of my two basis states system I write down a general state by choosing these amplitudes to be whatever you like to make any state of a whatsoever and let the state of b similarly be b up plus b down of down. Then what's the state? Now let's have a look at the state a, b. The state of a, b that we get well it's going to be this thing bracket into this thing a plus plus plus a minus minus b plus plus plus b oops sorry sorry this has the up and the down states b subscript down, ket down and when you multiply this out you get a disgusting mess because you get a plus plus sorry sorry a plus b up of plus up plus a plus b down of plus and down plus a minus b up of minus and up plus a minus b down of minus and down. So this state is now along it's now a linear combination of four states and this is strongly suggesting that these four states are basis states for the compound system and indeed we will show that they are times amplitudes which are these products of those individual amplitudes and these amplitudes have well-defined meanings so for example a minus b plus is the amplitude will be found minus and b what does I say up so take the mod square of this you get the probability that the experiment to measure a's property and b's property will be these particular values but what I'm trying to show is that this state is not the most general state and the way I'm going to do that is I'm going to calculate the probability that b let me do this the same way I've got it here that b is up given that a is in the plus state so this is the kind of this is a reasonable question we've measured and found that a is up and I now want to know suppose I measure b's property I found that a is plus will I find that b's property is up or down this is going to be the probability that given that I am where I am b will be found to be up ok well this is equal to the probability simply that we have up and the probability for being up and plus over the probability that a is plus now why is that if I would move this here then this would say that the probability of being up and plus is the probability of being plus times the probability that we get up given that we have plus this is a very important result from statistics this is classical probability theory this is known as Bayes theorem but it's really a trivial rearrangement of the rule for multiplying probabilities the probability to be up and plus is the probability for being plus times the probability if you are plus that you are up so this is nothing to do with quantum mechanics this is just a rule of probability theory which now plays a very important role in statistical inference in all the sciences physical and social alright so what is that that's the probability that we are up and plus over what this is having plus on a comes in we can have plus in a in two ways with a confidence system that is to be down or be up and they are mutually exclusive events so I can add their probabilities so this probability on the bottom is p up plus plus p down plus so what is this this is equal to 1 over dividing through 1 plus p down plus over p up plus what about this let's go back to that expression up there what's this probability p down plus p down whoops down and plus is equal to is equal to a plus b down and p up plus going up there p up plus is a plus b up so these a plus is cancel we need to take the mod square of this whole thing of course but the crucial thing is those things cancel so this is in fact equal to b down plus sorry this is equal to b this ratio so what's the point the point is that this probability is actually we've just shown it's independent of a plus and a minus so this probability does not depend on the state of a what does that mean physically heuristically it means that the systems are not correlated I've just calculated one specific conditional probability but you can calculate any other conditional probability and you'd find the same thing that the probability of any state of b is independent of what you assume about what the result of measuring a and so on these are uncorrelated systems so what we conclude from this is that when the state of a b is a product of a state of a times a state of b the systems are uncorrelated that's an important physical assumption now for example if you have a hydrogen atom is the location of the proton correlated with the location of the electron well of course it is the hydrogen atom is here you can be pretty damn certain the electron lies within a few nanometers within a nanometer of the proton if the proton is over here you can be pretty sure that the electron is within a nanometer of the proton it's over here the electron and the proton are very strongly correlated because they're physics there's a piece of Hamiltonian which is correlating them so we don't so we do expect systems to be correlated and that means we do not expect systems in general to have wave functions to look like to have states to look like that so let me see the point is that the I'm not going to go through the demonstration I think that I said let's go back up some way let's go back to let's go back to here so if these objects form a complete set of states of A and these objects form a complete set of states for B then it's not hard to persuade yourself that sorry that these objects form a complete set for AB alright so this is a complete set if these complete for their respective subsystems now what's this telling us this is telling us that any state of the system including correlated states which as I have tried to argue are natural states states in which the two subsystems are correlated they must be writable as linear combinations of these objects so the conclusion here is put that back and start over here so any state of AB can be written as AB equals the sum C, I, J summed over I summed over J of states A, I B J these states describe uncorrelated states in which the two subsystems are uncorrelated but this may be correlated probably is correlated so the way quantum mechanics introduces correlations between subsystems is by taking linear combinations of uncorrelated states we just had such a linear combination of uncorrelated states here and it turned out that in this case that was still an uncorrelated state because this was simply an expansion in terms of some basis states of a state which was already a product of just two states so the point is that the general state cannot be written this thing in general cannot be written like that even though when you see a long list of basis states it may with certain complex numbers in front it may be that the state can be written thus so whether this thing can be written as a product of two separate states depends on these numbers now we haven't got time to go into what property it is which ensures that you can do a decomposition like this into uncorrelated states which makes this state uncorrelated and when these are correlated but you can find a complete account of it in the book there are I think some and there are problems investigating this but the point is that if you in this concrete example here this is one of the seas another of the seas and these seas are not general they have the property you could arrange those in a 2x2 array of objects and if you this matrix of this 2x2 matrix is sort of a generic matrix it's a special matrix it's not the general one that you get by choosing these numbers independently so correlations go in like that and in quantum mechanics when you say that two states two systems are correlated you actually usually use the word entangled entangled is just the same thing it's just quantum mechanical jargon for correlated and what it means is if a compound system or two subsystems are entangled it means the state of the compound system cannot be written in that form it has to be written in this form and these numbers these numbers do not have the property that requires them that they have to have to enable them to be expressed as products of individual amplitudes of the individual systems so let's do a little bit of quick counting suppose there are M basis states of A and N of B right so there are N there are M values that I can take are then there are N values that J can take so then there'll be M times N amplitudes C i J so to specify a general state of the system you need to specify M N numbers C i J to specify a state but to specify A you need just M numbers A i and to specify B you need N amplitudes B J so to specify a general state of the form A B you need M plus N amplitudes so M plus N is generally much less than M N if you've got it in this little example M was 2 and N was 2 so this number was 4 and this number was 4 but supposing so they're the same but supposing that this number was 8 and this number was 8 then this would be 16 and that would be 64 so usually most systems are not two state systems usually so what this is telling us is that in a general state system there's very much more information than there is in here and why is that? because to specify a general state of the system you have to specify all the correlations between the subsystems and there are a lot of possible correlations this is not a problem only for quantum mechanics this would be a problem if we were doing statistical physics classical statistical physics correlations are nothing to do I mean not directly to do with quantum mechanics there are a logical problem there's a lot of inference also in the classical world and correlations are very hard to handle in classical probability theory they're actually easier in this apparatus here because quantum mechanics pulls this amazing trick correlated states of the system are obtained are understood as quantum interference some like this is a quantum interference between uncorrelated states of the system when you're doing classical probability theory you aren't able to pull that trick and it's much harder to specify correlations so correlations are important in both the classical word and the quantum world but they're actually easier to handle in the classical world because of the strange way in which quantum mechanics compounds these amplitudes does this quantum interference so quantum interference is how quantum mechanics handles correlations and because it has its own completely unique way of handling correlations the results can be surprising they can be ones that raise eyebrows and the Einstein-Badalski-Rosen experiment is an example let's try and pin these ideas a bit by looking at a concrete example of the H atom so in the position representation what do we want to know our complete set of amplitudes are going to be things like x so this is so let's make this the electron wave function and we're going to have we're going to have also so we'll call this xe and we will have xp times a big u this will be a proton wave function which gives you the amplitude to find the proton at the point xp this gives you the amplitude to find the electron at the point xe and supposing these things have subscripts on them, ui and uj so this might be the amplitude to find the electron so this might be an ui given that the energy of the electron is ei and this might be the amplitude to find the proton somewhere given that the proton's energy is ej say then what is the state a state of the H atom would be sorry xp xp so what is this? this is a state of the hydrogen atom in which the proton has this energy the electron has this energy and that gives me a state of the logically coupled pair of proton and electron this as I say is a very realistic state of the hydrogen atom because it's going to give us this says that the electron and the proton are uncorrelated and I've just tried to persuade you that the electron and the proton are very strongly correlated consequently their wave functions this isn't going to be a realistic useful wave function for hydrogen atoms as found in lab so what do we have to do a more realistic state might be xe xp shall we say chi for a new label which would be sum sum cij of xe ui xp uj but what are these? this is a boring function of x with a label i there's a set of functions of xe which have labels i and complex values and then this complex number is multiplied on this complex number which is a function of xp a member of a family of functions with labels j here is an amplitude another complex number at least complex number together and you get this complex number and this so any state of a hydrogen atom must be writable like this but realistic states are not writable like that because of this correlation of the proton and the electron ok now we need to revisit collapse oops of wave function so what happens when we make measurements on compound systems we know that when we make measurements what happens when we make measurements on a single system and we have to extend these ideas so suppose let's go back to our state of our system so we go back to the two state system two two state system a b and consider consider this particular state psi which is equal to a times plus up plus minus brackets b up plus c down supposing this is what we have this is pretty much written down at random it is a well defined state of the system because it's a sum of three of the four basis states that we were discussing it's a sum of plus up minus up and minus down this is the amplitude that if you would measure a and you would measure b you'd find that a was plus and b was up this is the amplitude for finding that a is minus and b is up etc but I've written this one down this state is as it turns out entangle that is to say you won't be able to write this as a product of a state a and a state b so this is more realistic than the states that I was discussing before ok ok now suppose suppose we measure so let's measure measure state subsystem a if we get plus then after measurement the theory says I'm not going to justify this I'm stating this as a conjecture which is equal to plus up so how does the system let's just remind ourselves what collapse of wave function was all about in the one state system in the one single system sorry if we get a single system we wrote up psi was equal to the sum an let's say en for example and we measured e the answer em then up psi went to the state m right after the measurement it was in this state so I'm making I'm stating that in this more complicated scenario where we have a two we have a composite system we measure only one of the subsystems we get a certain answer it goes to that state which is consistent with what we had over there because we found the answer plus so we threw away everything times minus but the whereas over there is simply em the coefficient up there of plus was not just a complex number a which was giving me the probability it was also times this state of b and this state of b just gets copied down so what does this say so this is what the theory claims is that that goes to that it doesn't explain how this happens this is the problem of measurement but there's a physical implication of this which is that you are now a measurement of b is guaranteed to produce or to find up right because this thing is something times up there is now zero amplitude to find down you are certain not to find down you are certain to find up if on the other hand we get minus for a then the new state is equal to minus sorry the new state is equal to minus brackets b up plus c down properly normalised so over the square root of b squared plus c squared so this is what the theory claims that if you get the minus thing then your new state is essentially the coefficient of minus and minus itself all properly normalised and now so if we get minus uncertainty as to what the result of a measurement on b will be so now measurement of b yields for example up with probability b squared over the square root of b squared plus c squared so we now apply the same old rules about the probability of measuring the interpretation of the amplitudes because we are certain to get minus if we measure if we measure a again but if we measure b we can get two outcomes either up or down and the probabilities are like that so that's a conjecture that's a statement, a theoretical statement about how the interpretation of the theory works and we just have to accept it and see whether it leads to proper experimental predictions so in our last minutes we have unfortunately a big topic to discuss which is operators for composite systems so we've talked exclusively so far about the kets but we know that operators play a very important role with every measurable quantity there's going to be an operator and we need to know how this behaves so we found that the kets of the subsystems were multiplied this rule was inherited from the multiplication of probabilities of successive events the operators so for example if we have two free particles if a and b are both free particles then ha is equal to pa squared the momentum of a squared over twice the mass of a and hb the Hamiltonian operator is equal to pb squared over to mb so what's the Hamiltonian of the combined system hab is equal to ha plus hb in other words it's pa squared over to ma plus pb squared over to mb and that's sort of saying the energies of the combined system is the sum of the energies of the individual bits does the operator we now need to explain how an operator pa operates on one of these states here so when pa hits a i b j so this is a state of the combined system and this is an operator which has to operate on the state of the combined system and what does it do it produces pa operating on ai which is a well-defined state of a symbolically times b j if pb works on this thing pb ignores this it passes through this as if pb was just an ordinary complex number and homes in on this it's target so this is simply ai times pb bj this is a well-defined state of b it gets to be symbolically multiplied by this well-defined state of a and there you are so for example what would the expectation value ab ij habb in this case here let's just make sure that we get some sense out of this sorry ab ij so what does that mean that means ai bi i sorry j ej brackets ha plus hb close brackets ai bj so this operator ignores that because it's a b operator and homes in on that this operator operates on this and then we have the other things coming on the other side and this gives me ai p sorry ha bi j bi j plus so that comes from this this and this because that passes through this a operator as if this was just a number and bangs into that plus correspondingly we're going to have ai bi j hb bi j this of course is going to be the number one this is going to be ea the expectation value of the energy of a this is the number one and this is the expectation value of b so we find that the expectation value of the energy of the combined system is lo and behold the sum of the energies of the individual bits I think that makes physical sense when the Hamiltonian takes that simple form if it's just the sum of the individual bits but for example for hydrogen the Hamiltonian h is equal to p electron squared over 2 mass of electron plus p proton squared over 2 the mass of a proton minus the charge on the electron squared over 4 pi epsilon 0 x electron minus x proton in modulus because the energy of the hydrogen atom is the sum of the kinetic energy of the electron and the kinetic energy of the proton and an interaction energy of the two because they electrostatically attract each other so this is equal to h electron plus h proton these being the Hamiltonians of the free electron and the free proton plus an interaction Hamiltonian and the thing about this interaction Hamiltonian is that it depends on operators belonging both to the first subsystem and the second subsystem and the consequence of that is that he comma h interaction commutator is not equal to 0 because the p e the electron momentum operator sitting inside here has a bone to pick with the electron position sitting inside here and similarly of course hp comma h interaction is not equal to 0 so without that interaction we would have that the so what's the important point about this is that the Hamiltonian of the hydrogen atom does not commute with the Hamiltonians of the electron and the proton you cannot know the energy so generically you do not expect to be able to know the energy of the hydrogen atom if you know the energy of the electron because they don't commute and it's the interaction that stops the computing well we're going to have to stop unfortunately at that point but we're pretty nearly done I'll just write down one final statement which is that the operators of different subsystems always commute so for example p proton comma x electron is precisely nothing et cetera we do not have to worry about non vanishing commutators of operators that belong to different subsystems okay